In this explainer, we will learn how to find the resultant of a group of forces acting at a point.
Suppose that multiple forces act at a point, such as in the figure below.
We call the net force exerted by the combination of all the forces the resultant force. In this instance, this is vector shown below.
To calculate the resultant of any number of forces, we can add the forces together tip to tail. That is, we move the starting point (or tail) of to the finishing point (or tip) of , we move the tail of to the tip of , and so on for however many forces we have. The result of this is shown below.
We note that this gives us a polygon of forces where the resultant force is the side formed by connecting the starting point to the tip of the final force, .
Another way to find the resultant is to add the perpendicular components of each force together. Suppose we have a force that can be decomposed into the horizontal and vertical components and , and similarly a force that can be decomposed into and . Then, we can find the resultant of these two forces by adding the components, as shown.
The perpendicular components of a force are labeled and , as shown in the following figure.
The magnitude of is given by where is the angle between and and is the angle between and .
, the component perpendicular to , is given by
The magnitude of the resultant of the perpendicular components of a force is given by
Let us look at an example of multiple forces acting at a point.
Example 1: Finding the Magnitude and Direction of the Resultant Force of Three Forces Acting on a Body
A body has a force of 10 newtons acting on it horizontally, 25 newtons acting on it vertically upward, and 5 newtons acting on it at an angle of to the horizontal as shown in the figure. What is the magnitude of the single resultant force acting on the body, and at what angle to the horizontal does it act? Give your answers correct to one decimal place.
Answer
The 5-newton force can be resolved into perpendicular components parallel to the - and -axes, as shown in the following figure.
The net force acting in the -direction is given by
The net force acting in the -direction is given by
and correspond to the two shorter legs of a right triangle, as shown in the following figure.
The length of the hypotenuse of this triangle equals the magnitude of the resultant force, which is, therefore, given by
Only the positive root of is used, as the magnitude of a force is necessarily positive.
To one decimal place, this is 31.6 newtons.
The angle from the horizontal at which the resultant force acts can be determined using the equation
We can use this formula to obtain :
To one decimal place, this is .
There is no limit to the number of forces that can act at a point. Each force that acts at a point can be resolved into perpendicular components. Let us look at an example of six forces acting at a point.
Example 2: Finding the Resultant of Six Forces Acting on a Regular Hexagon at a Point
The diagram shows a regular hexagon, , whose diagonals intersect at point . The 6 forces shown acting at are measured in newtons. Find , the magnitude of their resultant, and , the angle between their resultant and the positive -axis. Round your value of to the nearest minute if necessary.
Answer
The hexagon consists of 6 equilateral triangles, hence the angle between two forces that correspond to sides of any one of these triangles is .
Taking angles from the -axis clockwise, the net force acting in the -direction is given by
Recalling that the net force acting in the -direction is given by
and correspond to the two shorter legs of a right triangle. The length of the hypotenuse of this triangle equals the magnitude of the resultant force, which is, therefore, given by
Only the positive root of is used, as the magnitude of a force is necessarily positive.
The angle from the horizontal at which the resultant force acts can be determined using the equation
To the nearest minute, this is .
A force can be expressed in terms of its perpendicular components, where each component is expressed as the length of a unit vector in a direction perpendicular to that of the other component. Let us look at an example in which the resultant of forces expressed in such a way is determined.
Example 3: Finding the Magnitude and Direction of the Resultant of Three Forces Acting at a Point
The resultant of forces , , and makes an angle with the positive -axis. Determine , the magnitude of the resultant, and the value of .
Answer
The resultant of , , and in the -direction is the sum of the components of these forces, which is given by
The resultant of , , and in the -direction is the sum of the components of these forces, which is given by
The summed components of and are shown as force vectors in the following figure. The line from the tail of to the head of is the resultant, , of the components.
The magnitude of can be determined using the Pythagorean theorem:
The tangent of angle from the horizontal at which the resultant force acts can be determined using the formula
Now, let us look at an example in which the angles at which forces act are not directly given.
Example 4: Finding the Resultant of Four Forces Acting at a Point on a Triangle
is a triangle with a right angle at , where , , , and . Four forces of magnitudes 2, 3, 19, and 14 newtons are acting at point in the directions , , , and respectively. Find the magnitude of the resultant of these forces.
Answer
It is helpful to first draw to determine what this can reveal about the directions that the 19 N and 14 N forces act in. The following figure shows .
The tangent of is given by
Taking the sides of opposite and adjacent to as having lengths of and , respectively, the hypotenuse of has a length given by
From this, we see that and
The directions in which the forces act at are shown in the following figure.
From this, we see that the net force acting in the -direction, taking as positive, is given by
The net force acting in the -direction, taking as positive, is given by
The magnitude of the resultant force equals the magnitude of the net force in the -direction, which is 24.4 N.
Let us now look at an example in which the resultant of a set of forces is known but some of the forces contributing to the resultant are of unknown magnitude.
Example 5: Finding the Magnitudes of Two Unknown Forces out of a Group of Forces given the Free-Body Diagram
Forces of magnitudes , 16, , 18, and newtons act at a point in the directions shown on the diagram. Their resultant, , has a magnitude of 20 N. Find the values of and .
Answer
The angle between the horizontal and the 16 N force is given by
The net force acting in the -direction is given by
The net force acting in the -direction is given by
The resultant, , has a magnitude of 20 N. has a horizontal component equal to and a vertical component equal to . From this, we see that
We also see that
We can confirm that these values of and are correct by substituting these values into the expressions for and :
The magnitude of the resultant is given by which is the value of stated in the question.
Let us summarize what we have learned in these examples.
Key Points
- Multiple forces can be summed by summing the perpendicular components of the forces and determining the resultant of the components.
- If is one of the perpendicular components of a force , the magnitude of is given by where is the angle between and . , the component perpendicular to , is given by
- The magnitude of the resultant of the perpendicular components of a force is given by
- If a force is given by and a force is given by the resultant of these forces is given by
